| Abstract: |
| I will summarise the breakthrough result by Guillen and Silvestre (Acta Math, 2025) where they show the global well-posedness of the spatially homogeneous Landau-Coulomb equation. The Landau-Coulomb equation is a kinetic equation describing the statistical evolution of a plasma dominated by long-range Coulomb interactions. The result is based on proving that the Fisher information is monotonically decreasing along the solutions of this equation. They do so by considering a lifted equation where the dimension of the velocity space is doubled. In this lifted regime, the nonlinear degenerate Landau operator turns into a linear operator. They then prove the inequalities relating the Fisher information along the Landau operator and the Fisher information along the lifted operator which is studied by a special change of variables. Their result crucially relies on the symmetry assumption for the joint distribution of two colliding particles. Finally, I will present our recent results obtained in a collaboration with J. Junn\`{e} (TU Delft) and R. Winter (Cardiff) on the multi-species Landau equation where we construct more general Lyapunov functionals removing this symmetry requirement. |
|